Possible??

Magic always works.

Could you give some more details as to why the wireless game adapter is not an option. If cost is the issue, then you may be out of luck because most other options will be more expensive but one.

options:
1- powerline networking; uses existing wiring to network devices, plugs into outlets. $$$
2- phoneline networking; uses phone lines in house. $$$
3- run a ethernet hardline from xbox to router. This is cheapest option.
a) You can drill holes and run the wires in the walls,
b) run the wires in the heating vents (need plenum grade wire),
c) run the wires along the baseboards (sometimes you can tuck the wire under the baseboards),
d) or lastly, run a wire down the hall - stairs - thru the kitchen - living room to the xbox
4- move the xbox

Buffalo makes a wireless ethernet converter that is cheaper than most of the game converters on the market. Buffalo used to be widely accepted with some enthusiest, but less so nowdays.

If it wasn't the WRT54GC, you'd have some options with third-party firmware. And you could still do that if you want, taking the risk, with e.g. a Buffalo WHR-G54S and DD-WRT. This would be cheaper and more flexible in features than the off-the-shelf wireless client bridges such as the "Ethernet Converter", but the cost is the additional effort and risk you'd undertake installing third-party firmware.